Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is one of the generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is non-trivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is non-trivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In the present paper we construct structure-preserving integrators for a simple Nambu mechanical system, and derive the shadow Hamiltonians in two ways. This is the first attempt to derive shadow Hamiltonians of structure-preserving integrators for Nambu mechanics. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role to calculate the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

翻译：辛积分器是哈密顿力学中广泛使用的数值积分器，它保持系统的哈密顿结构（辛性）。尽管辛积分器不守恒系统能量，但众所周知存在一个守恒的修正哈密顿量，称为影子哈密顿量。对于作为广义哈密顿力学之一的Nambu力学，我们也可采用与构造辛积分器相同的步骤构造保结构积分器。然而，在保结构积分器中，影子哈密顿量的存在性并非显然的。这是因为Nambu力学由多个哈密顿量驱动，且积分器的时间演化能否归结为多个影子哈密顿量驱动的Nambu力学时间演化是一个非平凡问题。本文针对一个简单的Nambu力学系统构造保结构积分器，并通过两种方法推导影子哈密顿量。这是首次尝试推导Nambu力学保结构积分器的影子哈密顿量。我们证明，利用Baker-Campbell-Hausdorff公式计算影子哈密顿量时，对应于哈密顿力学中Jacobi恒等式的基本恒等式起着关键作用。结果表明，所得影子哈密顿量根据基本恒等式的使用方式不同而呈现不定形式。这不是技术假象，因为独立获得的精确影子哈密顿量同样具有这种不定性。